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Max–min algebra is defined as a linearly ordered set with two binary operations. Classical addition and multiplication are replaced by maximum and minimum, respectively. A square matrix is called robust, if all its orbits converge to an eigenvector. In practice, matrix inputs and start vector inputs are rather contained in some intervals than exact values. Considering matrices and vectors with interval coefficients is therefore of great practical importance. In this paper, we study a special type of the robustness called the X -robustness, the case when a starting vector is limited by a lower bound vector x _ and an upper bound vector x ¯ . If we consider some components of the interval starting vector with the existential quantifier and others with the general quantifier, two new types of X -robustness arise, called X E A -robustness and X A E -robustness. Using a similar quantification of interval matrix elements, we obtain AE/EA- X E A -robustness and AE/EA- X A E -robustness. A complete characterization of AE/EA- X A E -robustness and AE/EA- X E A -robustness for a special type of interval matrices, so-called interval circulant matrices, is presented. [ABSTRACT FROM AUTHOR]

In max-min algebra the standard pair of operations plus and times is replaced by the pair of operations maximum and minimum, respectively. A max-min matrix A is called strongly robust if the orbit x;A x;A2 x;: :: reaches the greatest eigenvector with any starting vector. We study a special type of the strong robustness called the strong X-robustness, the case that a starting vector is limited by a lower bound vector and an upper bound vector. The equivalent condition for the strong X-robustness is introduced and efficient algorithms for verifying the strong X-robustness is described. The strong X-robustness of a max-min matrix is extended to interval vectors X and interval matrices A using for-all-exists quantification of their interval and matrix entries. A complete characterization of AE/EA strong X-robustness of interval circulant matrices is presented. [ABSTRACT FROM AUTHOR]

In max-min fuzzy algebra, the study of eigenvectors is important because it can help us to recognize the steady states of systems working in discrete steps. This paper investigates the properties of steady states described by max-min matrices and vectors with interval coefficients. The characteristics of the eigenspace structure and polynomial-time algorithms for recognition of tolerable and weakly-tolerable interval eigenvectors in max-min algebra are described. This research is a continuation of an earlier investigation concerning strongly-tolerable interval eigenvectors. [ABSTRACT FROM AUTHOR]

Max-min algebra is defined as a linearly ordered set with two binary operations. Classical addition and multiplication are replaced by maximum and minimum, respectively. A square matrix is robust, if its eigenspace is achieved starting at arbitrary vector and it is X -robust if its eigenspace is achieved starting at each vector from a given interval vector X . The EA and AE robustness and X -robustness of interval circulant matrices over max-min algebra are defined. Polynomial algorithms for checking these types of robustness and X -robustness are given. [ABSTRACT FROM AUTHOR]

Max-min algebra is an algebraic structure in which classical addition and multiplication are replaced by maximum and minimum, respectively. The notation A ⊗ X ⊗ C = B , where A , B , and C are given interval matrices, represents an interval max-min matrix equation. The paper deals with the solvability of interval matrix equations in max-min algebra. We define three types of solvability of interval max-min matrix equations, namely the strongly universal, universal and weakly universal solvability. We provide the equivalent conditions for each type of solvability that can be verified in polynomial times. [ABSTRACT FROM AUTHOR]

In max-min algebra the standard pair of operations: plus and times is substituted by pair of operations ⊕ and ⊗, where a ⊕ b = max ⁡ { a , b } , a ⊗ b = min ⁡ { a , b }. A square matrix A is X − robust if its eigenspace is reached starting at each vector of a given interval vector X = { x ; x _ ≤ x ≤ x ‾ }. Various versions of the reachability for an interval vector X and an interval matrix A = { A ; A _ ≤ A ≤ A ‾ } depending on the used quantifiers and their order are studied. The universal, possible, tolerance and weak tolerance A -robustness of X and X -robustness of A over max-min algebra are defined and polynomial algorithms for its checking in the case of circulant matrices are described. [ABSTRACT FROM AUTHOR]

In max–min fuzzy algebra, the standard pair of operations—plus and times—is substituted by a different pair of operations—maximum and minimum—which are involved in many optimization problems. Investigation of the properties of eigenvectors is important for these applications. The values of vector or matrix inputs in practice are usually not exact numbers and some intervals can instead be considered as values. This paper investigates the properties of matrices and vectors with interval coefficients. In addition, a complete solution of the strong tolerance interval eigenproblem in max–min algebra is presented. • The eigenproblem in max–min algebra for matrices and vectors with interval coefficients is investigated. • The strong tolerance interval eigenproblem has been completely solved. • Preserving the interval security level in computer net with data storage units and logical units is described. [ABSTRACT FROM AUTHOR]

A matrix A is said to have X-simple image eigenspace if any eigenvector x belonging to the interval X = fx: x ≤ x ≤ xg containing a constant vector is the unique solution of the system A y = x in X. The main result of this paper is an extension of X-simplicity to interval max-min matrix A = fA: A ≤ A ≤ Ag distinguishing two possibilities, that at least one matrix or all matrices from a given interval have X-simple image eigenspace. X-simplicity of interval matrices in max-min algebra are studied and equivalent conditions for interval matrices which have X-simple image eigenspace are presented. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras. [ABSTRACT FROM AUTHOR]

An n ×n interval matrix A = [A, A] is called to be a fuzzy interval matrix if 0 ≤ Aij ≤ Aij ≤ 1 for all 1 ≤ i, j ≤ n. In this paper, we proposed the notion of max-min algebra of fuzzy interval matrices. We show that the max-min powers of a fuzzy interval matrix either converge or oscillate with a finite period. Conditions for limiting behavior of powers of a fuzzy interval matrix are established. Some properties of fuzzy interval matrices in max-min algebra are derived. Necessary and sufficient conditions for the powers of a fuzzy interval matrices in max-min algebra to be nilpotent are proposed as well. [ABSTRACT FROM AUTHOR]

Abstract: Interval eigenvectors of interval matrices in max–min algebra are investigated. The characterization of interval eigenvectors which has been presented in [7] for increasing eigenvectors, is extended here to general interval eigenvectors. Classification types of general interval eigenvectors are studied and characterization of all possible six types is presented. [Copyright &y& Elsevier]

In this paper, the minimal solutions of max–min fuzzy relational equations are investigated. A sufficient and necessary condition, for discriminating whether a given solution is minimal or not, is shown. Furthermore, we propose a new algorithm for computing all minimal solutions less than or equal to a given one. [Copyright &y& Elsevier]

Abstract: A speed-up of a known O(n 3) algorithm computing the period of a periodic orbit in max–min algebra is presented. If the critical components (or the transitive closure A +) of the transition matrix A are known, the computational complexity of the algorithm is O(n 2). This is achieved by using only those coordinates of the orbit that are related to the critical components. On the other hand, no critical component can be omitted. As the critical components are pairwise disjoint, the new formula is helpful also in solving the converse problem: generating an orbit with a prescribed period. This is demonstrated by examples. All parts of the paper are connected with the fact that the periodic regime of an orbit is encoded in its critical coordinates. We show that the non-critical coordinates of the state vector after n −1 steps can be ignored, how to generate a known periodic regime by using a small number of coordinates of the state vector and that the difference between the defect of an orbit and the quasidefect of its critical coordinates is small. The final part deals with the situation when the matrix is reduced after the orbit has reached its periodic regime. [Copyright &y& Elsevier]

Abstract: For a given linear mapping, determined by a square matrix in a max–min algebra, the set consisting of all vectors with a unique pre-image (in short: the simple image set of ) is considered. It is shown that if the matrix is generally trapezoidal, then the closure of is a subset of the set of all eigenvectors of . In the general case, there is a permutation , such that the closure of is a subset of the set of all eigenvectors permuted by . The simple image set of the matrix square and the topological aspects of the problem are also described. [Copyright &y& Elsevier]

The problem of the strong regularity of a square matrix in a general max–min algebra is considered and a necessary and sufficient condition using the trapezoidal property is described. The results are valid without any restrictions on the underlying max–min algebra, concerning the density, or the boundedness. Previous results on this topic are special cases of the theorems presented in this paper. [Copyright &y& Elsevier]

The problem of the strong regularity for square matrices over a general max–min algebra is considered. An O(n2logn) algorithm for recognition of the strong regularity of a given n×n matrix is proposed. The algorithm works without any restrictions on the underlying max–min algebra, concerning the density, or the boundedness. [Copyright &y& Elsevier]